I agree with the other ladies re asking for referral to a specialist Uro-Gynae unit. Medics who deal with this problem are usually v sympathetic & used to dealing with ladies of menopausal age (so no need to be embarrassed). As already mentioned, referral to a women's pelvic floor physiotherapist often follows & can be invaluable, ensuring you're doing the exercises correctly to get the most out of them. In the meantime, you may find it helpful to have a look at pelvicphysio.com. It's important to learn to relax the pelvic floor muscles as well as how to contract effectively, as inadequate relaxation cause urinary retention (incomplete emptying) resulting in the frequent need to pass often unsatisfying, small volumes. In these circs UTIs become more likely and can lead to interstitial cystitis.
Like Michelemabelle I use a muscle toning device but for a different condition; not specifically designed to help with bladder/PF probs, but for abdominal muscle tone. However, I've found this too can tighten the pelvic floor on the day of use to the extent that it hinders emptying. Took me a while to put 2 & 2 together, but I've found reducing the session to around 10 mins is less likely to cause probs.
I have to remind myself from time to time that alcohol (sorry!) fruit juice, caffeine & anything acidic can irritate the bladder, leading to annoyingly frequent urge to empty. When we're busy it's also easy to forget to drink enough, with the result that harmful bacteria are more likely to build up when not so frequently flushed out.
Anything resulting in swelling of the vulva/urethral area such as thrush or BV can also make bladder control trickier & these conditions may go unnoticed if mild in the context of increased discomfort due to dryness/atrophy.
I agree with the other ladies about the value of topical oestrogen, but would just add that I often find my bladder feels irritated the day following application, though this soon settles and is a small price to pay for keeping the entire uro-genital area in better condition for the long term.
The bottom line is that help is available, so please don't be afraid to push for it. Good luck with the GP.